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Abstract

Although perfectly matched layers (PMLs) have been widely used to truncate numerical simulations of electromagnetism and other wave equations, we point out important cases in which a PML fails to be reflectionless even in the limit of infinite resolution. In particular, the underlying coordinate-stretching idea behind PML breaks down in photonic crystals and in other structures where the material is not an analytic function in the direction perpendicular to the boundary, leading to substantial reflections. The alternative is an adiabatic absorber, in which reflections are made negligible by gradually increasing the material absorption at the boundaries, similar to a common strategy to combat discretization reflections in PMLs. We demonstrate the fundamental connection between such reflections and the smoothness of the absorption profile via coupled-mode theory, and show how to obtain higher-order and even exponential vanishing of the reflection with absorber thickness (although further work remains in optimizing the constant factor).

Figures (8)

(a) PML is still reflectionless for inhomogeneous media such as waveguides that are homogeneous in the direction perpenendicular to the PML. (b, c) PML is no longer reflectionless when the dielectric function is discontinuous (non-analytic) in the direction perpendicular to the PML, as in a photonic crystal (b) or a waveguide entering the PML at an angle (c).

Reflection coefficient as a function of discretization resolution for both a uniform medium and a periodic medium with PML and non-PML absorbing boundaries (insets). For the periodic medium, PML fails to be reflectionless even in the limit of high resolution, and does no better than a non-PML absorber. Inset: reflection as a function of absorber thickness L for fixed resolution ~ 50pixels/λ : as the absorber becomes thicker and the absorption is turned on more gradually, reflection goes to zero via the adiabatic theorem; PML for the uniform medium only improves the constant factor.

Reflectivity vs. pPML thickness L for the 1d periodic medium (inset) with period a, as in Fig. 2, at a resolution of 50pixels/a with a wavevector kx = 0.9π/a (vacuum wavelength λ = 0.9597a, just below the first gap) for various shape functions s(u) ranging from linear [s(u) = u] to quintic [s(u) = u5]. For reference, the corresponding asymptotic power laws are shown as dashed lines.

Field convergence factor [Eq. (11)] (~ reflection/L2) vs. pPML thickness L for the discontinuous 2d periodic medium (left inset: square lattice of square air holes in ε = 12) with period a, at a resolution of 10pixels/a with a vacuum wavelength λ = 0.6667a (not in a band gap) for various shape functions s(u) ranging from linear [s(u) = u] to quintic [s(u) = u5]. For reference, the corresponding asymptotic power laws are shown as dashed lines. Right inset: ℜ[Ez] field pattern for the (point) source at the origin (blue/white/red = positive/zero/negative).

Reflectivity vs. PML thickness L for 1d vacuum (blue circles) at a resolution of 50pixels/λ, and for pPML thickness L in the 1d periodic medium of Fig. 4 (red squares) with period a at a resolution of 50pixels/a with a wavevector kx = 0.9π/a (vacuum wavelength λ = 0.9597a. In both cases, a C∞ (infinitely differentiable) shape function s(u) = e1-1/u for u > 0 is used, leading to asymptotic convergence as e-α√L for some constants α.

Reflectivity vs. PML thickness L for 1d vacuum (inset) at a resolution of 50pixels/λ for s(u) =u2, with the round-trip reflection either set to R0 = 10-16 (upper blue line) or set to match the estimated transition reflection from Fig. 3 (lower red line). By matching the round-trip reflection R0 to the estimated transition reflection, one can obtain a substantial reduction in the constant factor of the total reflection, although the asymptotic power law is only changed by a lnL factor